This paper considers the development of adaptive beamforming algorithms for a prototype volumetric array. Volumetric arrays are desirable in passive sonar systems because of their ability to resolve left-right ambiguity. However, adaptively beamforming volumetric arrays poses several unique challenges compared to linear arrays such as the large number of hydrophones, lack of sample support, the necessity to know accurately the x,y,z positions of all the elements, and the correlated structure of the ambient noise, even when it is isotropic. These problems are addressed by exploiting the special geometry of the prototype volumetric array by first adaptively forming beams with each of the three individual line subarrays using an expanding Krylov space obtained from a vector conjugate gradient algorithm and then adaptively beamforming the reduced-dimension line subarray beam outputs with Capon's method. The new approach is demonstrated on actual at-sea data and compared against conventional adaptive beamforming methods and is shown to work well and be highly robust.